The Ricci Curvature and the Normalized Ricci Flow on Generalized Wallach Spaces

Abstract

We proved that the normalized Ricci flow does not preserve the positivity of the Ricci curvature of invariant Riemannian metrics on every generalized Wallach space with \(a_1+a_2+a_3\le 1/2\), in particular, such a property takes place on the homogeneous spaces \(\operatorname {SU}(k+l+m)/\operatorname {S}(\operatorname {U}(k)\times \operatorname {U}(l) \times \operatorname {U}(m))\) and \(\operatorname {Sp}(k+l+m)/\operatorname {Sp}(k)\times \operatorname {Sp}(l) \times \operatorname {Sp}(m)\) independently on their parameters k, l and m. We proved that the positivity of the Ricci curvature is preserved under the normalized Ricci flow on generalized Wallach spaces with \(a_1+a_2+a_3> 1/2\) if the conditions \(4\left( a_j+a_k\right) ^2\ge (1-2a_i)(1+2a_i)^{-1}\) are satisfied for all \(\{i,j,k\}=\{1,2,3\}\). We also established that the homogeneous spaces \(\operatorname {SO}(k+l+m)/\operatorname {SO}(k)\times \operatorname {SO}(l)\times \operatorname {SO}(m)\) satisfy the above conditions if \(\max \{k,l,m\}\le 11\), moreover, additional conditions were found to keep \(\operatorname {Ric}>0\) in cases when \(\max \{k,l,m\}\le 11\) is violated. Answers have also been found to similar questions about maintaining or non-maintaining the positivity of the Ricci curvature on all other generalized Wallach spaces given in the classification of Yu. G. Nikonorov.